Music Informatics Alan Smaill March 3, 2016 Alan Smaill Music - - PowerPoint PPT Presentation

T H E U N I V E R S I T Y O F E D I N B U R G H

Music Informatics

Alan Smaill March 3, 2016

Alan Smaill Music Informatics March 3, 2016 1/22

T H E U N I V E R S I T Y O F E D I N B U R G H

Today

Different harmonic organisation Pitch classes and their transformations Computers and Composition?

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Harmonic organisation

So far we have mostly considered harmony from WTM, where there are standard notions of key, cadence and so on. We saw that paradigmatic analysis can function without building in assumptions that are present in Lerdahl and Jackendoff’s GTTM, for example. There are many other ways in which pitch-space can be organised; today mostly look at the atonal or serial music which abandoned notions of key at the start of the last century, and then looked for

• ther ways to organise harmony.

Some of the ways of describing harmonies here were developed in 1960s influenced by the early use of computers.

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Forte on Atonal Music

Allen Forte’s book “The Structure of Atonal Music” gives a general theory that analyses this sort of music using the notion of pitch-class sets (introduced earlier by Babbitt). There is an on-line account of the basic ideas, with an applet illustrating some of the operations, due to Jay Tomlin, at http://www.jaytomlin.com/music/settheory/help.html and a useful summary of the theory from Paul Nelson: http://composertools.com/Theory/PCSets/

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Pitch-class set

We are working here in a situation of equal temperament, where there is no pitch distinction between c♯ and d♭. In a pitch-class set, pitches are taken modulo the octave, so octave displacements are

• ignored. The following example is due to Forte.

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Compare

Compare ending of piece by Sch¨

• nberg (the final chord below,

from The Book of the Hanging Gardens, Op 15, no 1) to the start

• f piece by Webern (6 Pieces for Orchestra Op 6, No 3):
• p
• sf
• pp
• Now move by notes to fit pitches in the smallest possible range.

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Comparison

• So regard these chords as (transposed versions of) the same

pitch-class set. There is some controversy as to whether this sort of relation can be consciously heard by listeners; but this does indicate some sort

• f similarity; and we can say more about this in this case.

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Representing pitch-class sets

Since we don’t care about transpositions, or the order of notes, then there is a simple way to describe pitch-class sets. Number the pitches from 0 to 11 (0 for c natural, 11 for b natural). Just give set of integers – traditionally here, write [0,2,5]. It turns out we can get a unique representation by finding fitting notes into smallest range by doing octave transpositions, and transposing so that the bottom note is C (=0); however if this does not uniquely specify the representation: choose lowest difference between first two entries if this does not uniquely specify the representation: choose lowest difference between first and third entries . . .

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Relations between pc sets

The unique resulting representation is called the normal form. Some operations relate sets together, in a way similar to operations in music using strict counterpoint. Transposition: as in the example above, take intervals in the same direction: [C,E,G] ⇒ [F, A, C]; using shared pitch classes get: [0, 4, 7] ⇒ [5, 9, 12] and these have the same normal form.

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Inversion

Another relation betweem pitch classes is inversion: Inversion: take intervals in the opposite direction: [C,E,G] ⇒ [C, A♭, F]; after getting normal representation if the sets, get: [0, 4, 7] ⇒ [0, 3, 7] Two pitch sets are called equivalent iff they have the same normal form, or the normal form of one is the normal form of the inversion

• f the other.

The prime form of a set is its normal form, or the normal form of its inversion, whichever is smaller. (see Jay Tomlin’s page). Thus the major/minor distinction is abolished (in that the chords are equivalent).

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Complement

Another relation is that between a pc set and its complement: the pitch-classes that are not in the given pitch class. So [C,D,E♭, F, G, A] ⇒ [C♯,E,F♯,G♯,B♭,B] or [0, 2, 3, 5, 7, 9] ⇒ [0, 2, 4, 6, 7, 9] Can we hear this? Probably not – but the literal case, where no transposition or inversion is involved, does have the effect of filling up the space of semi-tones. (The version with note names is this literal case, pc sets are not.)

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Subset, superset

There is also the relation between 2 pc sets, where one has takes

• nly some of the pitch classes from the other – making it a subset.

We know the literal version of this from classical harmony, where the major triad is a subset of the chord with the flattened 7th added. Note that if we use unique representations, it might not be obvious which set is a subset of another, since transposition and inversion are allowed. eg, [0, 1, 6] ⊂ [0, 2, 3, 8] These ideas are influenced by the mathematical notion of set, as was Xenakis in his book “Formalized Music: Thought and Mathematics in Composition”.

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So what?

In Forte’s work, these ideas are part of a way of analysing music: – look for relations between the pitches in groups of notes. This takes us back to a familiar issue: segment the musical surface (into small segments here), and then compare pitch classes, or use pitch classes to recognise segmentation When done by hand, there is a mixture of both processes.

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Pitch class set complexes

A further idea is the a bunch of pitch class sets can be related together in a complex, e.g. K*(T) the collection of the sets that are subsets of a given set T; K(T) the collection of sets that are subsets or subsets of the complement of the given set T; Kh(T) the collection of sets that are subsets and subsets of the complement of the given set T.

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Schoenberg Op 19, no 6

The handout shows the score for this very short piano piece, and also some parts of Forte’s analysis. The music has echoes of romantic harmony, but does not fit into WTM tradition for overall rhythmic or harmonic structure (Forte, “The Structure of Atonal Music”, pp 97, 98). Still, it does not at all sound random to the listener.

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Schoenberg ctd

Forte’s analysis gives a way of understanding the relationships between parts of the musical material. It uses Forte’s own terminology such as names for the unique pc set representations which are here attached to the score. The details are not so important here; what is claimed is that this abstract representation relating pitches explains some of the ways in which this piece works as music (as far as pitch organisation is concerned). It acts as a replacement for classical notions of major/minor/dimished chords, their inversions, and classical harmonic progressions.

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Use of computer in analysis

Using a computer to find and/or verify such analyses has benefits: It is easy to get things wrong working by hand, by claiming relations which are or, or (more likely) missing relations which hold. It forces the analyst to be precise about the relations involved, and what exactly is being looked for. It is much easier to see how analysis can be implemented on the basis of this theory than say using GTTM.

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Implementation

paper on basic capabilities:

http://music.columbia.edu/~akira/JDubiel/ PaperOnJDubielICMC09.pdf

and applied to George Perle’s very similar ideas, from Computer Music Journal

http://www.mitpressjournals.org/doi/pdf/10.1162/comj.2006. 30.3.53

In the latter case, as in the L&J approach to grouping, there is still a large amount of ambiguity, and a good tool will allow an analyst to steer the way through to a preferred reading of segmentation on the basis of 12-tone pitch relationships.

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What current implementations can do

Look at end of CMJ paper: We have demonstrated in the present work a program that can compute useful information given a sequence of chord segments, performing in seconds what could take weeks or months for a human analyst. The natural question to ask is how much more can be automated, and what tasks absolutely require the expertise of a human analyst. . . .

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Current implementations ctd

When working with segments of fixed sizes (three to six in the current work), the number of possible arrangements is small enough to deal with. However, if an algorithm is given an entire composition, there are so many possible ways to segment it that it would not be possible to do an exhaustive analysis, even with a

• computer. We must find some heuristics . . .

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For the composer

These ideas have also been an explicit part of the compositional process for some composers. Can the computer help here too? From CMJ article: the composer may enter input from a variety of perspectives, including potential pitch class segments, or desired interval cycles . . . and the application will provide as output a wealth of information for the composers

• consideration. In short, the program can supply the

composer with a comprehensive set of pre-compositional material, based on the composers specifications. As such, then, the T3RevEng program is a practical, robust tool for both analysts and composers working within the parameters of twelve-tone tonality.

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Summary

Pitch classes as an abstract representation of pitch

• rganisation (particularly in atonal music)

Computer assistance using these abstractions for analysis and composition.

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